減振鏜桿的結(jié)構(gòu)設(shè)計外文翻譯

M,彼天z探埠院本科生畢業(yè)設(shè)計（論文）外文翻譯原文標題Active regenerative chatter suppression during boring manufacturing process譯文標題鏜削再生顫振的積極抑制作者所在系別機械工程系作者所在專業(yè)xxxxx作者所在班級xxxx作者姓名xxxx作者學號xxxx指導(dǎo)教師姓名xxxx指導(dǎo)教師職稱講師完成時間2012 年 2 月北華航天工業(yè)學院教務(wù)處制譯文標題鎮(zhèn)削再生顫振的積極抑制原文標題Active regenerative chatter suppression during boring manufacturing process作 者Chiang C.Mei 譯 名 赤昂.西梅 國籍 英國原文出處Robotics and Computer-Integerated Manufactering鏜削再生顫振的積極抑制 摘要鏜削的加工性能經(jīng)常受加工中振動的影響在各種振動來源中，再生顫振是最不 利的它不僅限制了切削深度，對表面質(zhì)量也有不利的影響，同時也會損害工具壽命 盡管加工系統(tǒng)是一種分布式的系統(tǒng)，通用的控制器是根據(jù)一個簡化后的單自由度切削 過程模型來設(shè)計的。

這是因為大部分切削過程只存在著一個主導(dǎo)模式然而，簡化后 就會出現(xiàn)一些問題首先，因為系統(tǒng)本身是分布式系統(tǒng)，理論上它是由無數(shù)個振動模 型組成，當控制器僅僅控制主要的模型，被用來控制主導(dǎo)模型的能量會激起原本靜止 的機構(gòu)的振動，即引起所謂的溢流問題第二、單自由度控制器設(shè)計的成功依賴于有 效的精確的模型參數(shù)（如質(zhì)量當量，剛度，阻尼），但不幸的是獲取這些參數(shù)非常困 難因為所有的參數(shù)隨加工過程的變化而變化在這篇論文里，將從波的觀點，在廣 義的頻率中來吸取振動能來設(shè)計減振器，來提高鏜刀的加工性能，相對于當前流行的 控制器設(shè)計，波控制器的設(shè)計是真正基于分布式切削系統(tǒng)模型，這個抑制顫振的控制 模式的主要優(yōu)點是它的穩(wěn)定性這個控制模式也可以消除控制溢流問題一、簡介金屬切削中的振動總體上是受迫振動和自激振動引起的受迫振動是由回轉(zhuǎn)件的 失衡引起的，比如失衡的驅(qū)動系統(tǒng)，伺服不穩(wěn)定或者多齒零件的撞擊受迫振動可以 認為是由振動頻率和受迫力頻率對比引起的，但相應(yīng)的措施可以被用來減小或消除這 些振動來源自激振動包括兩部分：基本類型（不可再生類型）和再生類型不可再生式自激 振動出現(xiàn)在回轉(zhuǎn)件的波動表面對系統(tǒng)的振動沒有相影響的時候，如車螺紋。

因此它只 和切削過程中受到的力有關(guān)再生式自激振動是因為工具通過時，系統(tǒng)的振動和回轉(zhuǎn) 件的波動表面相互作用產(chǎn)生的，再生式自激振動對加工過程的影響最為不利，因此它 是這篇論文的焦點再生顫振抑制已經(jīng)被做了大量的研究，傳統(tǒng)設(shè)計中阻尼和剛度是被謹慎處理的 被動和主動的減振器已將開始被應(yīng)用了，我們可以根據(jù)動力學的波形圖來選擇軸的速 度和速度變化主動控制器采用了速度反饋，最佳的和最適合的算法盡管加工系統(tǒng)是分布式的， 但所有通用的主動控制器都是根據(jù)單自由度的切削模型來設(shè)計的這樣的簡化存在一 些潛在的問題首先，因為系統(tǒng)本身是分布式系統(tǒng)，理論上它是由無數(shù)個振動模型組 成，當控制器僅僅控制主要的模型，那被用來抑制主要模型的能量會激起原本靜止的 模型的振動，這就不可避免的引起了所謂的溢流問題單自由度控制器設(shè)計的 成功依賴于有效而精確的模型參數(shù)(如質(zhì)量當量，剛度，阻尼)，但不幸的是獲取這 些參數(shù)非常困難因為所有的參數(shù)隨金屬切削和機床移動部件的變化而變化這篇論文的目的在于通過一種反饋波控制器來解決以上在實際分布式切削系統(tǒng) 中存在的問題這種控制實際思想不僅可以消除控制溢流，也能大大的減小主要模型 和剩余的分布式模型引起的振動。

此外，反饋波控制器的設(shè)計還要求有結(jié)構(gòu)局部特征 方面的知識，它對系統(tǒng)的不確定因素敏感度很小，因此更加穩(wěn)定二、系統(tǒng)模型加工系統(tǒng)通?？梢员荒M成帶有邊界范圍的一維分布式結(jié)構(gòu)對于一個不回轉(zhuǎn)的 鏜削加工，工件比鏜刀本身還硬典型鏜刀的扭轉(zhuǎn)硬度比彎曲硬度要大，因此它可以 被模擬成一種懸臂梁使用一維分布式結(jié)構(gòu)的彎曲歐拉梁，動力學平衡方程式如下：ei ^xn+p/2 心七)=f(x, t) ⑴辦 4 dt 2x是沿著橫梁的距離，t為時間，f (x,t)為集中力，E為彈性模量，p為密度，A 為截面面積古典偏微分方程的解決方案通常使用本征函數(shù)展開響應(yīng)通過無數(shù)個組件模式之 和來表達：w(x,t)=黨 4 (x)q (t) (2)i=1七(x)和qi(t) (i=1,2, ?8)分別是系統(tǒng)的模態(tài)形狀和模態(tài)坐標這種模態(tài)形狀 通常使之標準化：jL pA@. (x)巾.(x)dx = 8 , i, j = 1,2,...8, (3)L是懸臂梁的長度，(5是狄拉克函數(shù)運動的方程式可以用模態(tài)坐標來描述：q (t) + w2q (t) = f (t),i = 1,2...,a ⑷f是模態(tài)力：if (t) = jL (x,t)4 (x)dx, i = 1,2,...,a. (5)1 0 1應(yīng)該指出，在實際中式組件的數(shù)量通常用n替代無窮。

在一個主導(dǎo)模式的情況下，n =1.在切削力和刀具在切削過后留下的起伏不定的工件表面相互作用下，方程式1變?yōu)椋篸 4 w(x, t) . d 2 w(x, t)EI + pA + k B(w(t) - ^w(t - T)) = f (x, t), (6a)kc是切削剛度，由工件材料和刀具形狀決定的，B是切削深度，T是周期，p是 重疊因子，p在0到1之間變化，在最壞的情況下(p =1)，上式變?yōu)椋?4 w(x, t) . - 2 w(x, t)EI + pA + k B(w(t) - w(t - T)) = f (x, t), (6b)運用模態(tài)坐標來表達動力方程式，和方程1到4相同的步驟：q (t) + w2q (t) + bw2(q (t) 一 q (t - T) = f (t), i = 1,2..., s. (7)i i i i i i i'b= kb / EI是關(guān)于切削深度的無量綱參數(shù)，w,?是第i個模式下的自然頻率方程式7是結(jié)構(gòu)無阻尼自由振動微分方程，事實上因為沒有純粹的結(jié)構(gòu)無阻尼自 由振動微分方程存在，假設(shè)結(jié)構(gòu)存在粘滯阻尼，方程式就變?yōu)椋簈 (t) + 2& w q (t) + w2(q (t) + bw2(q (t) 一 q (t - T)) = f (t),. (8)i ' ' ' ' ' ^' ^' ,' '通常&?.是個0到1間很小的正數(shù)，最常用的是&^0.05.三、反饋波控制的設(shè)計這種金屬切削過程中的振動抑制反饋控制是從波的觀點來設(shè)計的。

當遇到一個障 礙物，入射的振動波一部分是透射，一部分是反射主動控制力就是被用來產(chǎn)生這樣 一個“主動的”障礙物，以便于來控制透射和反射的振動波，考慮一個搭配的傳感器 和執(zhí)行機構(gòu)的安排，假設(shè)波控制力作用于一點，控制力在頻率范圍內(nèi)可以寫成以下方 程式：F =-H (w)W (9)Hw (w)是控制器的頻率響應(yīng)傳輸矩陣t和反射矩陣r通過考慮力作用點的連續(xù)性和平衡性條件獲得的t=I+p c, r=p c (10)I是單位矩陣i i H (w) / 、c = , h = , (11)1 1 w 3/2 -(1 + i) H (w)'H (w) = H (w) / 4<'(pA) 3 EI .w一系列可能的控制方法在反饋波控制器的設(shè)計過程中被研究這里提供一個最佳的能量吸收控制器，被設(shè)計來最大化的吸收在一個寬頻帶里由起伏的部件引起的振動 能也就是讓|,|2 +陣最小因此最佳的控制器被認為是：H(w) = a4(pA)3 EI (1 + i)w3/2. (12)注意理想的控制器具有頻率依賴性，在實際執(zhí)行中，控制器必須接近于理想在 通常金屬切削過程中，只存在一個主模式因此這個控制器可以通過調(diào)節(jié)控制器在主 模式的頻率內(nèi)達到最佳來獲得。

這接近于最佳的控制器是：H (w) = c + c (iw), (13)其中 c = 24(pA)3Eiwd3/2 and c2 = 2&pA)3EIwj假設(shè)波控制力被應(yīng)用于七點，控制力相對于變量t的方程式可寫為：f (x, t) = -[c w(x, t) + c w(x, t)]6 (x - x ). (14)它可以看成比例和導(dǎo)數(shù)調(diào)節(jié)器控制增益的設(shè)計也是從從波的觀點出發(fā)的實現(xiàn)了波控制后方程式(6b)變成：d4 w(x, t) . d2 w( x, t)EI + pA —初2 + k B(w(t) - w(t - T)) = f (x, t) + f (x, t),dx 4(15)運動方程式可以再一次分解:寫成矩陣形式:q (t) + 2& w q (t) + w2 q (t) + bw2 (q (t) - q (t - T)) +1 i i i i i i i ic 工巾(x 沖(x )q (t) + c 工巾(x 沖(x )q (t) = f (t)2 i c i c i i j c i c j ij=1 j=1(16)M q(t) + C q(t) + Kq(t) + Kq(t - T) = f (t),q = [q (t)q (t)?：q (t) ]t , f = [ f (t) f (t)...f (t)]t ,1 2 n 1 2 nC = A + c ? (x 沖(x ), K = (1 + b)A + c ? (x 沖(x ),ij ij 2 ij c j c ij ij 1 i c jKd =-bA「.ij 2 ij c j c ij其中M=I,矩陣上T表示向量的裝置。

引入狀態(tài)向量X(t)=T.qT : q(17),方程式17(18)可寫為:X (t) = AX (t) + At X (t - T) + Bf (t)與系數(shù)矩陣0-M -1K1-M -1C, 0 0 ,At = - M -1K 0and,BL d 」0M -1四、穩(wěn)定性高的金屬切削率和高的表面質(zhì)量無疑是加工中最重要的問題然而合適的切削率 建立在機床系統(tǒng)的動力學基礎(chǔ)上，已知的加工系統(tǒng)存在一個不穩(wěn)定的臨界值，波形圖 是用來描述臨界值和允許的最大切削深度的，刀具的性能和刀具/工件的旋轉(zhuǎn)速度之 間的關(guān)系在加工系統(tǒng)的研究中波形圖是一個非常有用的工具由方程式18,得特征方程式：sI - A - A e一st = 0. (19)T波形圖是通過設(shè)置方程式19里的S-來解特征方程，導(dǎo)致最大的允許切削深度和 無窮小參數(shù)B有關(guān)，在給定切削速度的情況下根據(jù)以上研究，在切削過程中只存在一個主要的模式(解釋了為什么主動控制器 的設(shè)計是基于一個自由度的離散型模型)因此，在以下數(shù)字實例中，我們假定只有 一個主要模型，即N=1圖1是沒有控制的情況下工件系統(tǒng)的波形圖，圖2是力作用于0.5L的情況下工 件系統(tǒng)的波形圖鏜刀被認為是彈性模量E = 180GN/m2，密度p = 8600mg/m3.的鋼 桿。

桿的直徑為2英寸，長徑比為10.結(jié)構(gòu)的阻尼系數(shù)為0.01.在系統(tǒng)的主要模式下 控制器將是最佳的固有頻率七假定為791.5rad/s.因此 匕=2 x 791.53/2,七=2 x 791.51/2，從圖1和圖2可以看出切削深度在有控制后得到顯著 地提高0.20.060.040.020：OCO 20DO 3000主軸就400050006000圖1.未控制時的系統(tǒng)穩(wěn)定性0.2° 0 1000 2000 3000 4000 5000 60DQ主軸錘.圖2.控制后景統(tǒng)的穩(wěn)定性需要指出的是金屬切削過程中的動力是唯一的，系統(tǒng)的參數(shù)在切削過程中不斷的 變化，盡管單自由度模式聽起來和在一個主要模式情況下的分散式系統(tǒng)非常接近，但 它卻需要連續(xù)的模式參數(shù)才能適當?shù)墓ぷ鞑恍业氖且郧暗难芯繘]有一份提到這個問 題反饋波控制器是基于機構(gòu)本身性能在一個寬頻范圍內(nèi)來最佳的吸收振動能的原理 來設(shè)計的，控制器時穩(wěn)定的隨著系統(tǒng)參數(shù)的改變時間的時域響應(yīng)通過q (頃=1,2,...)來獲得，然后把他們帶到方程式2中采用庫 i塔法和拉格朗日插值定理解決微分問題，方程變?yōu)槿缦拢? (x) = cosh k x — cos k x — b (sinh k x — sin k x), (20)i i i i i i固有頻率和波數(shù)的關(guān)系式如下w = k2\；EA.i i以為只有一個主模式，響應(yīng)可以近似描述為：w(x, t) =8 (x)q (t), (21)時間圖4,系曲制后的時間響應(yīng)I4 1（x）由方程式20給出，七=1.875/L （L為棒料的長度），。

1= 0.7341比如，從圖1可以看出在4400rpm時，無量綱的切削深度b為0.01，當增加b到 0.04時根據(jù)波形圖會發(fā)生不穩(wěn)定的振動，圖3中的時間響應(yīng)已經(jīng)證實了這個響應(yīng)發(fā) 生在刀具的工件的接觸點上，假定是懸臂梁的頂部在安裝主動控制器后，在4400rpm無量綱的切削深度b為0.04時的不穩(wěn)定切削 得到了控制圖4的時間響應(yīng)證實了這一發(fā)現(xiàn)響應(yīng)圖說明了反饋波控制很打的提高 了允許的切削深度，在軸的速度給定情況下隨著切削深度增加，生產(chǎn)力也得到了提 高五、結(jié)論在這篇論文里，一個從波的觀點出發(fā)在一定頻率范圍內(nèi)吸收振動能的減振器被應(yīng) 用在再生振動的鏜削過程中盡管在系統(tǒng)中有一個主要的模式主導(dǎo)響應(yīng)，但剩下的模 式也或多或少影響著系統(tǒng)的振動此外，當控制器僅僅由主模式來設(shè)計時，就可能會 出現(xiàn)控制溢流（被用來控制主要模式的力會引起剩余模式的變化），就像從勃朗寧實 驗觀察的一樣在這次研究中，控制器在在很寬的頻率內(nèi)設(shè)計的，阻尼在這頻率范圍 內(nèi)的模式上因此它不僅能消除溢流問題，也能很好地減小主模式和其他模式引起的 振動原文：active regenerative chatter suppression during boring manufacturingprocessAbstractMachining performance such as that of the boring process is often limited by chatter vibration at the tool-workpiece interface. Among various sources of chatter, regenerative chatter in cutting systems is found to be the most detrimental. It limits cutting depth (as a result, productivity), adversely affects surface finish and causes premature tool failure. Though the machining system is a distributed system, all current active controllers have been designed based upon a simplified lumped single degree of freedom cutting process model. This is because it was found that in the majority of cutting processes, there exists only one dominating mode. However, such simplification does have some potential problems. First, since the system itself is a distributed system, theoretically it consists of infinite number of vibration modes. When the controller is designed to control the dominating mode(s)1. IntroductionChatter in metal cutting process, in general, is the result of both forced and self-excited vibrations.Forced vibration is due to the unbalance of rotating members, such as unbalanced driving system, a servo instability, or impacts from a multi-tooth cutter. In practice, the forced vibration sources can be traced by comparing the frequency of chatter with the frequency of the possible force functions. Corresponding measures can then be taken to reduce/eliminate such vibration sources.Self-excited vibration consists of two types, namely primary (or non-regenerative type) and regenerative type. The primary/non-regenerative type of self-excited vibration occurs when theses is no interaction between the vibratory motion of the system and the adulatory surface produced in the revolution of the work piece, such as that in threading. Hence if is inherently related to the dynamics of the cutting process. While the regenerative type of self-excited vibration is due to the interaction of the cutting force and the work piece surface undulations produced by previous tool passes. The regenerative type of self-excited vibration is found to be the most detrimental phenomena in most machining process, hence it becomes the focus of this paper.A considerable amount of work has been done on regenerative chatter suppression. Conventional structural damping and stiffness treatments are considered. Both passive and active vibration absorbers are implemented. Based on the dynamics of stability lobe pockets, spindle speed variation and spindle speed selection strategies are proposed.Active controllers are also designed using rate feedback, optimal and adaptive algorithms. Though the machining system is a distributed system, all current active controllers were designed based upon a simplified lumped single degree of freedom cutting process model. This is because it was found that in the majority of cutting process, there exists only one dominating mode. However, such simplification does have some potential problems. First, since the system itself is a distributed system, theoretically it consists of infinite number of vibration modes. When the control is designed to control only the dominating mode(s), the energy designed to suppress the particular mode(s) may excite the rest of the structural modes, This can causes a spillover problem. Second, the success of the designed controller of a simplified single degree of freedom system relies on the availability of accurate model parameters (such as the effective mass, stiffness and damping), which is unfortunately very hard to acquire. This is not only because theCutting process itself is a dynamic process but because the cutting process itself is a dynamic process but also because the system involves movable components of the machine tool and the work piece .This paper aims at solving the above mentioned problems by implementing a feedback wave controller on the real distributed cutting system model. Such a control design strategy not only eliminates control spillover, but also greatly reduces chatter caused by the dominating mode as well as that by the rest of the design of the feedback wave control requires only knowledge with regard to the local properties of the structure. It is thus much less sensitive to global system’s uncertainties, as a result, more robust.2. System modelMachining systems, in general, can be modeled as one-dimensional distributed structures with various boundary conditions. For a non-rotating boring process, the work piece is much stiffer than the boring bar itself. And typically boring bars are much stiffer in torsion than in bending. Hence it can be modeled as a cantilevered rod in bending. Using Euler-Bernoulli ’s bending model for one-dimensional uniform distributed structure, the equation of motion is as follows:84w(x, t) a 2w(x, t)EI + P A = f (x, t), (1)8x4 at 2where x is the distance along the beam, t the time, f(x,t) the external force, E the Young ’s modulus, I the cross-sectional area moment of inertia, p the mass density and A the area of cross section of the boring bar.The solution to the classical partial differential equation is usually found using Eigen function expansions. The response can be expressed as a sum of an infinite number of modal components asw(x, t) = u n. (x)q (t) (2)i=1Where n (x) and q (t) (i=1,2, ???8)are the mode shapes and modal coordinates of the system , respectively. The mode shapes are in general, mass-normalized such thatjL pAn. (x)n. (x)dx = 8 i, j = 1,2,...8, (3)where L is the length of the structure and。

the Dirac delta function.The equations of motion can be written in terms of the model coordinates as..q (t) + w2q (t) = f (t), i = 1,2..., 8. (4)i .. .where f is the i th modal force given byif (t) = jL (x, t)n (x)dx, i = 1,2,..., 8. (5)i 0 1]It should be pointed out that in practice. Only a finite number of modes are excited, as a result, the number of modal components is in general n instead of 8 .In the case of one dominating mode, n=1.Now considering the dynamic interaction of the cutting force and the workpiece surface undulations produced by previous tool passes during the cutting process, Eq.(1) then becomes [1]d4 w(x, t) . d2 w(x, t)EI + pA + k B(w(t) - pw(t - T)) = f (x, t), (6a)Where k is cutting stiffness determined by workpiece material and tool geometry, B the depth of cut and T the tooth passing period, p is the so-called overlap factor, which accounts for the overlapping of successive cuts. The value of varies between 0 and 1. Considering the worst-case scenario (wherep =1), then the above equation becomes-4 w(x, t) . - 2 w(x, t)EI + pA + k B(w(t) - w(t - T)) = f (x, t), (6b)The corresponding equations of motion in terms of the modal coordinates, by following the same procedures as that of Eqs.(1)-(4),are..q (t) + w2q (t) + bw2(q (t) 一 q (t - T) = f (t), i = 1,2..., 8. (7)i ' i ’ / / ’Where b= k b / EI is the cutting depth-related dimensionless parameter and w the natural frequency of the I i th mode.Eq.(7) describes an undammed structure. Since no purely undammed structures exist inreality, assuming viscous damping in the structure, the equation of motion is then rewritten as(8)q (t) + 2& w q (t) + w2 (q (t) + bw2 (q (t) 一 q (t - T)) = f (t),.. ... i i i i i iiUsually & is a small positive number between 0 and 1, with most common values of3. Feedback wave control designThe vibration suppression feedback control of the metal cutting process is designed from wave standpoint. Upon meeting a discontinuity an incident vibration wave is partially transmitted and partially reflected. An active control force is applied to create such an ‘a(chǎn)ctive’ discontinuity; so as to control the transmitted and/or reflected vibration waves. Considering a collocated sensor and actuator arrangement and assuming the wave control force being a point force, the control force can be written in frequency domain asF =-H (w)W , (9)Where H (w) is frequency response of the controller.The transmission and reflection matrices t and r are obtained considering the continuity and equilibrium conditions at the force applied pointt=I+p c, r印 c, (10)Where I is the identity matrix and(11)H (w)w 3/2 -(1 + i) H (w)Where H (w) = H (w)/4^ (p4)3 EI .wA number of possible control strategies have been studied in feedback wave control design. Here an optimal energy absorbing controller is adopted, in which the controller is designed to maximize the absorbed vibration energy caused by the propagating wave components in a broad frequency band, in other words to minimize \r2 + |t|2. In the case the optimal controller is found to beH(w) = a<'(p4)3 EI (1 + i)w3/2. (12)Note that the ideal controller is frequency dependent and non-causal. For a practical implementation, a causal controller must be found which an approximation to the ideal [29] is. It is found generally that in metal cutting processes there exists only one be obtained by turning the controller to be optimal at the corresponding natural frequency of the dominating mode w , the approximated optimal controller is thenH (w) = c + c (iw), (13)Where 匕=2不pA)3Eiw3/2 and c2 = 2&pA)3EIwjAssuming the point wave control force is applied at xc, the control force in time domain is given byf (x, t) = -[c w(x, t) + c W(x, t)]6 (x - x ). (14)It can be seen that it corresponds to proportional and derivative controller. Here the control gains are designed, however, from wave standpoint.After the implementation of wave control, Eq.(6b) becomesd4 w(x, t) . d2 w(x, t)EI—8 4 + pA—初2 + k B(w(t) - w(t - T)) = f (x, t) + f (x, t), (15)The equation of motion can again be decomposed in terms of the uncontrolled modes and becomesq (t) + 2& w q (t) + w2 q (t) + bw2(q (t) - q (t - T)) +i i i i i i i i iV v (16)c 乙巾(x )4 (x )q (t) + c 乙巾(x )4 (x )q (t) = f (t)2 i c i c i i j c i c j ij=1 j=1Which can be written in matrix form asM q(t) + C q(t) + Kq (t) + K q (t - T) = f (t), (17)Whereq = [q (t)q (t)...q (t) ]t , f = [ f (t) f (t)...f (t)]t ,1 2 n 1 2 nC = A + c 4 (x )4 (x ), K = (1 + b)A + c 4 (x )4 (x ),ij ij 2 ij c j c ij ij 1 i c j c% =-bA「.In which M=I, the identity matrixA = diag(w2 )andi, j = 1,2,...,n. iSuperscript T denotes the transpose of vector.TIntroducing state vector X(t) = qT : q , Eq.[17]can be rewritten in state-space fromasX(t) = AX (t) + A.X (t - T) + Bf (t) (18)With the coefficient matrices1-M -iC-M -i Kd0M -i4. StabilityHigh metal removal rate with quality surface finish is undoubtedly the most important issue in machining. While the applicable metal removal rate depends on the dynamics of the machine tool system, an instability threshold exists for a given machining system. Stability charts (or lobes) describe the relationship among the stability threshold, the relationship among the stability threshold, the maximum allowable cutting depth, the tool dynamics and the rotational speed of the tool/workpiece. The stability lobe diagram has been an important tool in the study of chatter in machining systems.From Eq.(18), the characteristic equation is obtained as(19)sI - A - A^e-st = 0.The stability lobe is obtained by setting s in Eq.(19) to jw and solving the characteristic equation, which result in the maximum allowable cutting depth related dimensionless parameter b at a given tooth passing speed .According to previous studies, there, in general, exists only one dominating mode in most metal cutting process (which explains the reason that almost all the active control design was based on a single degree of freedom discrete model). Hence in the numerical example considered below, it is assumed that the number of dominating mode is one , that is , n=1.Fig.1 shows the stability lobe of the uncontrolled tool system and Fig.2 shows the stability lobe of the controlled tool system with the control force applied at 0.5L. The boring bar is assumed to be a steel bar with Young’s modulus E = 180GN / m 2and mass density p = 8600mg / m3. The diameter of the rod is 2 inches with slenderness ratio being 10. The damping ratio of the structure is assumed to be 0.01. The controller is tuned to be optimal at the system ‘s dominating mode, that is the first natural frequency w ,which is calculated as 791.5rad/s. Accordingly, one has c = 2 x 791.53/2 and, c = 2 x 791.51/2 .From Figs.1 and 2 it can be seen that the cutting depth is improved greatly after control.It should be point out that the dynamics of metal cutting process is unique in the way that the system’s parameters change dynamically during metal removal process. Though the previous lumped single degree of free model is theoretically sound in approximating the distributed system in the case of one dominating mode, it demands continuous model parameters identification and updating to work appropriately. Unfortunately, none of the previous studies addressed this issue. With the feedback wave controller designed based on the local properties of the structure to optimally absorb vibration energy in a broad frequency band, the controller is robust to changes in system’s parameters.The time domain response is obtained by finding the model states q (t)(i = 1,2,...) numerically and then substituting them into Eq.(2). The adaptive Runge-Kutta-Ferlberg method [31] is used to solve the differential equations numerically with the delay term calculated using the Lagrange interpolation theorem.The mode shapes of a cantilever are given as8 (x) = cosh k x — cos k x — b (sinh k x — sin k x), (20)i i i i i iWhere k L = —~^^, b =，訕匕乙，詛匕，an^, j -1,2, A, 8 The natural frequencies i 2 i cos k L + cos k Land the wave numbers are related by w = k2 X;EI / pA.In the case that only one mode is dominating, the res。